A283535
a(n) = Sum_{d|n} d^(3*d + 1).
Original entry on oeis.org
1, 129, 59050, 67108993, 152587890626, 609359740069674, 3909821048582988050, 37778931862957228818561, 523347633027360537213570571, 10000000000000000000152587890754, 255476698618765889551019445759400442, 8505622499821102144576132293474637113130
Offset: 1
a(6) = 1^(3+1) + 2^(6+1) + 3^(9+1) + 6^(18+1) = 609359740069674.
Cf. Sum_{d|n} d^(k*d+1):
A283498 (k=1),
A283533 (k=2), this sequence (k=3).
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f[n_] := Block[{d = Divisors[n]}, Total[d^(3 d + 1)]]; Array[f, 12] (* Robert G. Wilson v, Mar 10 2017 *)
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a(n) = sumdiv(n, d, d^(3*d+1)); \\ Michel Marcus, Mar 11 2017
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N=20; x='x+O('x^N); Vec(x*deriv(-log(prod(k=1, N, (1-x^k)^k^(3*k))))) \\ Seiichi Manyama, Jun 18 2019
A308698
Square array A(n,k), n >= 1, k >= 0, read by antidiagonals, where A(n,k) is Sum_{d|n} d^(k*d).
Original entry on oeis.org
1, 1, 2, 1, 5, 2, 1, 17, 28, 3, 1, 65, 730, 261, 2, 1, 257, 19684, 65553, 3126, 4, 1, 1025, 531442, 16777281, 9765626, 46688, 2, 1, 4097, 14348908, 4294967553, 30517578126, 2176783082, 823544, 4, 1, 16385, 387420490, 1099511628801, 95367431640626, 101559956688164, 678223072850, 16777477, 3
Offset: 1
Square array begins:
1, 1, 1, 1, 1, ...
2, 5, 17, 65, 257, ...
2, 28, 730, 19684, 531442, ...
3, 261, 65553, 16777281, 4294967553, ...
2, 3126, 9765626, 30517578126, 95367431640626, ...
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T[n_, k_] := DivisorSum[n, #^(k*#) &]; Table[T[k, n - k], {n, 1, 9}, {k, 1, n}] // Flatten (* Amiram Eldar, May 09 2021 *)
A308757
a(n) = Sum_{d|n} d^(3*(d-2)).
Original entry on oeis.org
1, 2, 28, 4098, 1953126, 2176782365, 4747561509944, 18014398509486082, 109418989131512359237, 1000000000000000001953127, 13109994191499930367061460372, 237376313799769806328952468217885, 5756130429098929077956071497934208654
Offset: 1
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a[n_] := DivisorSum[n, #^(3*(# - 2)) &]; Array[a, 13] (* Amiram Eldar, May 08 2021 *)
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{a(n) = sumdiv(n, d, d^(3*(d-2)))}
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N=20; x='x+O('x^N); Vec(x*deriv(-log(prod(k=1, N, (1-x^k)^k^(3*k-7)))))
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N=20; x='x+O('x^N); Vec(sum(k=1, N, k^(3*(k-2))*x^k/(1-x^k)))
Showing 1-3 of 3 results.